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首页Linear Algebra Foundations to Frontiers
Linear Algebra: Foundations to Frontiers (LAFF) is an experiment in a number of different dimensions. • It is a resource that integrates a text, a large number of videos (more than 270 by last count), and hands-on activities. • It connects hand calculations, mathematical abstractions, and computer programming. • In encourages you to develop the mathematical theory of linear algebra by posing questions rather than outright stating theorems and their proofs. • It introduces you to the frontier of linear algebra software development. Our hope is that this will enable you to master all the standard topics that are taught in a typical introductory undergraduate linear algebra course.
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Copyright © 2014, 2015, 2016 by Margaret E. Myers, Pierce M. van de Geijn, and Robert A. van de
Geijn.
10 9 8 7 6 5 4 3 2 1
All rights reserved. No part of this book may be reproduced, stored, or transmitted in any manner
without the written permission of the publisher. For information, contact any of the authors.
No warranties, express or implied, are made by the publisher, authors, and their employers that the
programs contained in this volume are free of error. They should not be relied on as the sole basis to
solve a problem whose incorrect solution could result in injury to person or property. If the programs
are employed in such a manner, it is at the user’s own risk and the publisher, authors, and their
employers disclaim all liability for such misuse.
Trademarked names may be used in this book without the inclusion of a trademark symbol. These
names are used in an editorial context only; no infringement of trademark is intended.
Library of Congress Cataloging-in-Publication Data not yet available
Draft Edition,
This “Draft Edition” allows this material to be used while we sort out through what mechanism we
will publish the book.
Contents
0. Getting Started 1
0.1. Opening Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.1.1. Welcome to LAFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.1.2. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.1.3. What You Will Learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2. How to LAFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.1. How to Navigate LAFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.2. Setting Up to LAFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.3. Software to LAFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.3.1. Why MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.3.2. Installing MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.3.3. MATLAB Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.4. Enrichments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.4.1. The Origins of MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.5. Wrap Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.5.1. Additional Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.5.2. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1. Vectors in Linear Algebra 7
1.1. Opening Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1. Take Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2. Outline Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3. What You Will Learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2. What is a Vector? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2. Unit Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3. Simple Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1. Equality (=), Assignment (:=), and Copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2. Vector Addition (ADD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3. Scaling (SCAL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.4. Vector Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4. Advanced Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1. Scaled Vector Addition (AXPY) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2. Linear Combinations of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.3. Dot or Inner Product (DOT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.4. Vector Length (NORM2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.5. Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.6. Vector Functions that Map a Vector to a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5. LAFF Package Development: Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.5.1. Starting the Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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